Composite Structures 135 (2016) 306–315

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Composite Structures

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A comparative study of twill weave reinforced composites undertension–tension fatigue loading: Experiments and meso-modelling

http://dx.doi.org/10.1016/j.compstruct.2015.09.0050263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: xujian1028@hotmail.com (J. Xu).

J. Xu a,⇑, S.V. Lomov a, I. Verpoest a, S. Daggumati b, W. Van Paepegemb, J. Degrieck b

aKU Leuven, Department of Materials Engineering, Kasteelpark Arenberg 44, B-3001 Leuven, BelgiumbGhent University, Dept. of Materials Science and Engineering, Technologiepark-Zwijnaarde 903, 9052 Zwijnaarde, Belgium

a r t i c l e i n f o

Article history:Available online 30 September 2015

Keywords:Meso-scaleTextile compositeFatigue damageFinite elements

a b s t r a c t

The tension–tension fatigue characteristics of two types of twill weave carbon/epoxy composite materi-als have been experimentally investigated. The fatigue strain vs load cycle number is calculated with thehelp of digital image correlation (DIC). The increase of displacement amplitude and the degradation of thefatigue moduli for these two materials are studied and compared. The fatigue strengths obtained fromfatigue tests are used to compare with those predicted by the model and good agreements are obtained.The stress fields from meso-scale FE models are used to analyse the stress concentration that leads to thefibre rupture in the warp tows.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Woven composites are gaining popularity in the industry.However, as it is known, the crimp of the tows decreases the in-plane mechanical properties. As concluded in Ref. [1] damageand strength of the textile composite under axial loads are stronglyaffected by the fibre crimp. On the one hand, high flexure intro-duces severe local stress concentration, especially at the tow cross-over locations. On the other hand, the shear-bending interactionbetween warp and weft tows leads to an early inter-fibre matrixcracking in the tows transverse to the loads.

Many factors affect the mechanical behaviour of the wovencomposites. These factors include: (1) the mechanical propertiesof the fibre and matrix material, (2) the fibre/matrix interface, (3)the fibre volume fraction (VF) of the reinforcement materials, (4)the processing techniques (resin transfer moulding vs autoclave)and the processing parameters (curing time, temperature, pres-sure. . .), and (5) the reinforcement’s geometrical parameters suchas tow width, tow thickness, unit cell size and crimp.

Detailed literature reviews on fatigue analysis of textile com-posites have been provided by Degrieck and Van Paepegem [2],Post [3], Passipoularidis and Brøndsted [4] and Xu [5], respectively.According to Xu [5], the existing fatigue models can be classifiedinto three categories: Miner’s-rule-like models, phenomenologicalmodels and progressive damage models. In the recent, intensiveexperimental investigations have been carried out on fatigue

behaviour of woven composites [6,7], and comparative studieswere conducted by Carvelli et al. [8,9]. In Ref. [8] the non-crimpstitched UD and non-stitched UD are experimentally studied andthe stitching effects are analysed. The on-axis fatigue resistanceis enhanced by the structural stitching but weakened in theorthogonal direction. In Ref. [9], the 3D non-crimp orthogonalwoven composite and 2D plain weave laminate are comparativelyinvestigated. The conclusion is that the former composite has alonger fatigue lifetime and later damage onset. Nishikawa et al.[10] compared newly developed plain weave spread tow(cross-section aspect ratio = 20 mm:0.05 mm = 400) carbon/epoxyto conventional plain woven composites through static and fatiguetests. The spread tow composite showed an increase of staticstrength about 14% and 15 times prolongation of the fatigue life.A more comprehensive collection of the studies on textile compos-ites’ fatigue behaviour has been recently published [11] that coversthe experimental measurement, observation, numerical predictionand industrial application.

In this paper two types of twill weave carbon fibre epoxy com-posites are studied: (i) carbon/epoxy twill weave 3K tow laminates(CET3K) with (tow) width: thickness = 2 mm: 0.11 mm = 18, and(ii) carbon/epoxy twill weave 12K tow laminates (CET12K) with(tow) width: thickness = 5.45 mm: 0.16 mm = 34. The tow crimpsof these two impregnated fabrics are 0.85% and 0.28%, respectively.These two selected materials have the same fibre/resin system andfibre volume fraction, and are processed using autoclave followingthe same curing cycle, in order to avoid influences of theaforementioned factors (1)–(4). In author’s earlier work [1], thesetwo materials were experimentally and numerically characterised

J. Xu et al. / Composite Structures 135 (2016) 306–315 307

under static loadings. With higher crimp the CET3K exhibits earlierdamage initiation in weft tows and remarkably lower strengththan that of CET12K.

In the current work, the materials’ responses to fatigue loadingsare investigated through tension–tension fatigue tests and therecently proposed fatigue model [12]. The goal of the experimentsis to acquire the modulus-life curves and S–N plots. The test resultsof these two materials are compared and the different behaviour isanalysed. Meantime, the meso-FE model (Fig. 8) is used to quanti-tatively demonstrate stress concentrations that lead to fibre break-age and final failure, and differences in stress–strain fields for thetwo materials.

The impregnated tows are represented in the model as unidi-rectional composite (UD). In order to produce the S–N curves tofeed fatigue model, UD samples made from M10.1/T700S prepregsystem, the same as in CET3K (VF = 55.2 ± 0.08%) and CET12K(VF = 53.6 ± 0.17%), are tested under tension–tension fatigue load-ings, too. The M10.1 resin system that has high fatigue resistance issuitable for low pressure moulding processes allowing a range ofprocessing temperature from 85 �C up to 150 �C.

2. Tension–tension fatigue test

2.1. Experimental setup

The fatigue tests are conducted using MTS� 810 servo hydraulictesting machine equipped with a fatigue load cell of 100 KN andMTS� 647 Side-Loading hydraulic wedge grips. The operation sys-tem is TestStar-790.00 digital controller. A sinusoidal-waveconstant-amplitude tension–tension fatigue load with stress ratioR = 0.1 is applied to the samples. The tests are carried out in a con-ditioned room with a temperature of 18 �C. 3 Hz load frequency isuniformly applied to all tests. Two factors have to be consideredwhen defining the load frequency:

(A) The consistency of frequency for all samples: The frequen-cies of the fatigue loading applied to UD samples (S–N curvesto feed the model [12]) and textile composites are identical.In an ideal case the frequency has little effect on fatigue lifeof the composites when the temperature is strictly con-trolled [13].

(B) High level fatigue loadings: As high level fatigue loadings(75–85% of the static strengths – Table 1) will be applied,the load frequency should be less or equal than 3 Hz in orderto prevent the failure near to the end-tabs. De Baere et al.[13] often observed in-tab failures under fatigue loadings75% of the strength at 5 Hz frequency. He assumed higherfrequency leads to higher heat generation between tab andsample surface due to friction which decays the adhesiveand hence induces the in-tab failure. Lower than 3 Hz fre-quency would not be affordable in fatigue since the total testcampaign would be extended to more than one year [5].

2.2. Acquisition of the fatigue strain

All the fatigue tests in this work are load-controlled. However,the recorded grip displacement by the machine is not the trueelongation of the sample but the summation of the sample elonga-

Table 1Stress levels (%) and corresponding fatigue loadings (MPa).

CET3K CET12K

Static strength [1] 960 113275% 720 84980% 768 90685% 816 965

tion and the compliance of the test machine. Use of a dynamicextensometer was not possible because it tended to be detachedfrom the specimen in the very early test stage. Hence a relationbetween the grip displacement and the true strain in the middleof the sample was measured using optical extensometer (digitalimage correlation or DIC – Vic2D�) and then this relation appliedin fatigue testing to estimate the sample strain. The DIC averagingarea (80 mm � 25 mm) was set to the centre of the sample. The fullconfiguration was explained in Ref. [1].

In the first cycle a very low frequency (0.01 Hz) tensile load isapplied to sample, stretching the sample up to the maximum fati-gue load. The longitudinal strain, esample, and hence the sample’selongation, DsampleðFÞ can be obtained by Eq. (1):

DsampleðFÞ ¼ 2sample � Lsample ð1Þwhere Lsample is the sample free length. The total elongation, DtotalðFÞ,of the sample and machine is known from the MTS data file. Hencethe machine compliance, DmachineðFÞ vs load F can be obtained as:

DmachineðFÞ ¼ DtotalðFÞ � DsampleðFÞ; ð2Þas sketched in Fig. 1.

After removing the DIC, in the rest of the load cycles, the sam-ple’s elongation can be calibrated by Eq. (3):

DsampleðFÞ ¼ DtotalðFÞ � DmachineðFÞ ð3Þwhere DmachineðFÞ is the machine tension–deformation curveobtained from Eq. (2). The assumption is taken that the dynamiceffect (from 0.01 Hz to 3 Hz) will not give a relevant influence.Sample-grip slippage has to be strictly prohibited during all tests.Therefore a red mark was drawn on the grips aligning to the end-tabs to indicate any slippage.

2.3. Experimental data of carbon/epoxy UD

The M10.1/T700S prepreg system is processed into UD compos-ite plates by using autoclave as explained in detail in Ref. [5]. Thetest coupons are prepared following the description in Ref. [1]. Thewidth of the samples is 15 mm according to ASTM D3039 for UDcomposites. The mechanical properties are acquired through ten-sile tests using Instron� 4505 at a cross-head speed of 1 mm/mincombined with extensometer. The mechanical properties in thefibre direction are listed in Table 2.

The fatigue tests have been performed in the fibre direction.Four loading levels are predefined: 70%, 75%, 80% and 85% of thestatic strength. For each loading level, at least three valid testresults are necessary. Fig. 2(a) exhibits the normalised fatigue life

Fig. 1. Sketch showing how to calculate the machine compliance DmachineðFÞ vstensile loading by using DIC: the quasi-straight curve is obtained using DIC and thenon-linear one is based on the force–displacement data recorded by MTS�810.

Table 2Mechanical properties of UD composite through 3 valid test samples in fibre direction.

VF (%) Strength (MPa) Modulus (GPa)

55.3 ± 2.2 1345.1 ± 77.6 140.3 ± 7.2

308 J. Xu et al. / Composite Structures 135 (2016) 306–315

scatter. The Semi-Logarithmic-Bilinear regression model [14], asdiscussed in Ref. [12], is used to generate the S–N curve.

The Semi-Logarithmic-Bilinear model in Fig. 2(a) can be repre-sented by the following equations:

f ðNÞ ¼ 1:0 ðlog10N � 3:15Þf ðNÞ ¼ �0:09 � log10N þ 1:28 ð3:15 < log10N � 6:47Þf ðNÞ ¼ 0:70 ðlog10N > 6:47Þ

8><>: ð4Þ

In Fig. 2(a), the semi-logarithmic curve depicts that the fatiguestrength starts decreasing after one thousand cycles and reachesthe fatigue limit at about three million cycles. The material reachesfatigue limit (3 million cycles) at the load level about 70% of thestatic strength.

The on-axis catastrophic failure and the fatigue life are domi-nated by the fibre bundle rupture or on-axis S–N curve of UD interms of fatigue modelling. The S–N curve of UD composite inthe fibre direction will be used as the input data for the fatiguemodel to predict the on-axis fatigue life of the CET3K and CET12K.Further, the fatigue data of transverse fibre tension–tension fati-gue, in-plane shear fatigue and out-of-plane shear fatigue fromRef. [15] are utilised as input – Fig. 2(b)–(d). The material used in

Fig. 2. S–N curves of UD composite: (a) normalised S–N curve of UD composite under on-(b) transverse fibre tension–tension; (c) in-plane shear; (d) out-of-plane shear. Data in

Ref. [15] is AS4/3501-6 carbon/epoxy system, which may have dif-ferent matrix mechanical properties and interface properties com-pared to T700S/M10.1 system. Ref. [15] provides the mostcomplete test data and hence the best choice for the time being.However, for the fatigue behaviour in off-axis direction, wherethe matrix and fibre/matrix interface properties dominate thedamage progression, test samples using the same resin system asin the model are inevitable. Through FE-model, mild compressioneffects were observed in the weft yarn along the fibre direction,or perpendicular to the layer plane, due to Poisson’s ratio. Thesecompressive stresses are trivial and have, however, minor or nonecontribution to the material damage. Hence, for the time being, thecompressive loads are neglected.

2.4. Experimental results of twill composites

2.4.1. Displacement amplitude increaseThe tension–tension fatigue load is applied in warp direction of

the material. In a load-controlled fatigue test, the maximum andminimum of the applied load are constant throughout the test.Due to the (i) permanent deformation that is introduced bythermal stress release and fibre straightening and (ii) modulusdegradation, the maximum and minimum of the displacementresponse keep increasing during the fatigue life – Fig. 3. This phe-nomenon has been reported by Carvelli et al. [8]. Tested under aseries of fatigue loadings that have the same load frequency andload ratio, the maximum displacement (peaks) of the N-th cyclecan be presented as:

axis tension–tension fatigue: fatigue limit is 70% of the static strength at N = 3 � 106;(b)–(d) are extracted from Ref. [15].

Fig. 3. Sketch for the increase of the displacement peak (maximum), valley(minimum) and amplitude of the sample in fatigue test due to the materialdegradation and permanent deformation.

J. Xu et al. / Composite Structures 135 (2016) 306–315 309

PN ¼ P1 þ dp þ dPm ð5Þwhere P1, dp and dPm are the maximum displacement in the firstcycle, the permanent deformation and the deformation due to thereduction of the modulus under maximum load, respectively.Similarly, the minimum displacement (valleys) of the N-th cyclecan be presented as:

VN ¼ V1 þ dp þ dVm ð6Þwhere V1 and dVm are the minimum displacement in the first cycleand the additional elongation due to the reduction of the modulusunder minimum load. dVm is increasing throughout the fatigue lifebut quasi-constant in an arbitrary load cycle. Subtracting Eq. (6)from Eq. (5), it yields:

DA ¼ AN � A1 ¼ ðPN � VNÞ � ðP1 � V1Þ ¼ ðdPm � dVmÞ ð7Þwhere AN , A1 and DA are displacement amplitude of the Nth cycle,displacement amplitude of the first cycle and the increase of theamplitude (AI), respectively. Using Eq. (7) to process the experimen-tal data for CET3K and CET12K, the curves of DA for fatigue tests ondifferent load levels are shown in Fig. 4(a).

In order to show clearly the dependence of moduli’s reductionon material type, the test data in Fig. 4 are plotted based on thenormalised life cycle number. From Fig. 4(a), it can be observedthat: (i) the AI curves are clearly divided into two groups according

Fig. 4. (a) AI curves of CET3K and CET12K samples under fatigue load levels: 75%, 80% anfor CET3K and CET12K samples under fatigue loads of 75%, 80% and 85% of static streng

to material types; (ii) for CET12K, the ultimate AI is in a rangebetween 0.42 mm and 0.49 mm, higher than that of CET3K,between 0.15 mm and 0.20 mm; (iii) the ultimate AI tends to beirrelative to the load levels (either 75%, 80% or 85% of the staticstrength) but dependent on the material type.

2.4.2. Degradation of the sample stiffnessFig. 4(b) presents the normalised longitudinal fatigue moduli

decrease vs normalised fatigue life for CET3K and CET12K on threefatigue load levels. As well the curves are divided into two groupsbased on material types: CET3K samples have relatively moderatemodulus decreases through the tests, 13–17%, compared to thoseof CET12K, 23–30%. The higher applied loadings (Table 1) and thehigher modulus decreases of CET12K have clearly demonstrateda higher damage tolerance than CET3K, which has been suggestedby Fig. 4(a), too. Meantime, the averaged fatigue modulus decrease(15% for CET3K and 25% for CET12K) will be used in the fatiguemodel to determine the final fatigue failure as explained in Refs.[5,12].

Sharp drops of the moduli in the first and the last 5% of the fati-gue life are observed. This phenomenon was also reported forcross-ply UD composites in Ref. [16]. The first drop (stage I inFig. 4(b)) consists of transverse crack formation with increasingdensity, until a saturation state is reached. In stage II, delaminationoccurs with gradual breakage of those fibres which have defects.This stage lasts for about 90% of the fatigue life cycles at a muchlower and an approximately linear rate of modulus decrease. Inthe last drop (stage III), the moduli drop abruptly till the catas-trophic failure, formed by massive fibre breakage.

As seen from Fig. 4(b), stage II provides a major contribution (7%out of total loss 15% for CET3K and 17% out of total loss 25% forCET12K) to the modulus loss in fatigue. Through the morphologicalstudy shown in Fig. 5, it is found out that all the CET12K samples,failed at 85%, 80% and 75%, have developed severe fatigue damageaccompanied with fibre pull-out and tow splitting. In contrast,CET3K samples kept much better surface finishing. The much flat-ter reinforcement of CET12K may help the inter-laminar crackpropagation and, therefore, form the fatigue damage and delami-nation. The observed severer fatigue damage of CET12K gives onegood reason to the higher displacement AI and higher modulus lossof CET12K in fatigue.

2.4.3. Normalised modulus vs logarithmic cycle numberIn Fig. 6(a) and (b) the normalised moduli on different load

levels for CET3K and CET12K are plotted vs logarithmic life cycles.In these figures, the corresponding three stages of modulus degra-dation are also marked out for the stress level of 85%. Plotted on

d 85% of static strength; (b) Normalised modulus decrease vs normalised fatigue lifeth: three damage stages.

Fig. 5. The morphology of the samples after fatigue failure for CET12K and CET3K.They were fatigued under 85%, 80% and 75% of strength, respectively. CET12Ksamples show severe fatigue damage, fibre pull-out and rough surface.

310 J. Xu et al. / Composite Structures 135 (2016) 306–315

the logarithmic scale of the fatigue life, the moduli curves do notpresent the three stages as shown in Fig. 4(b). Stage I in Fig. 4(b)is stretched and shows slight decreases (less than 2%) before thefirst 1000 cycles. Right after 1000 cycles (until 10,000 cycles), themoduli start reducing drastically owing to the transverse crack for-mation that correlates to the first drop in Fig. 4(b). Stage II in Fig. 4(b) represents gradual degradation of the modulus, showing sharpdecrease in the logarithmic plot.

As expected, the degree of the modulus drop depends on loadlevels: the higher the load is, the faster the modulus reduces.

2.4.4. S–N plotThe fatigue strengths of CET3K and CET12K are normalised by

their static strengths, 960 MPa and 1132 MPa (Table 1), respec-tively. Three valid samples were tested for each considered loadlevel. The normalised strengths are plotted in Fig. 7(a) and (b).From these two figures, it is clear that (1) both of CET3K andCET12K reach fatigue limit (one million cycles) at 75% of the staticstrength and (2) they have about the same fatigue life under thesame normalised fatigue loading. When compared for the absolutevalues of the maximum fatigue stress in Fig. 7(c), however, themaximum fatigue stress corresponding to the same fatigue life isabout 18% higher for CET12K than that for CET3K, the same ratioas that of static strengths for these two materials.

The crimp is a typical structural feature and also a serious dis-advantage for a woven laminate that significantly reduces thestrength and introduces highly nonlinear response by high bending

Fig. 6. Normalised fatigue modulus vs logarithmic

torsion loading between crossed tows. Spread tow plain weavelaminate [17] with extremely low crimp has been experimentallyproven to have superior fatigue tolerance than conventional plainweave laminates. With lower crimp, CET12K (crimp 0.28%) hasproven its greater static and fatigue resistance over CET3K (crimp0.85%).

3. Progressive fatigue damage model

A meso-FE progressive damage model for textile compositesunder fatigue loadings has been recently published by authors[12]. This model is used to provide a quantitative study of the fati-gue damage as a consequence of stress concentration that is givenby tow interlock.

3.1. Methodology

This fatigue model is based on the mesoscale (unit cell) FEmethod – Fig. 8. A unit cell is the minimum repeating unit that pos-sesses the same properties as in the whole material. In term ofcommon textile composites, a unit cell (UC) occupies the same rep-resentative tow waviness, tow fibre volume fraction and tow geo-metrical parameters as in macroscale material. With the periodicboundary conditions (PBCs), the local strain and stress fields inthe full-scale material can be reproduced through a meso-FEmodel. In UC, the impregnated tows can be taken as piecewiseUD composites: the S–N curves of UD material will be used asinputs to predict the fatigue characteristics of a textile compositematerial – Fig. 8. The algorithm will be carried out by repeatingstep A and B:

(A) Quasi-static loading is applied to the unit cell as half cycle:Employing the static damagemodel [1], the unit cell is loadedup to the magnitude of the maximum fatigue stress rmax.Some Gaussian points are identified as failure and their stiff-ness is degraded following the anisotropic damagemodel [1].The degraded properties will be passed to Module B.

(B) Material is ‘worn out’ due to increasing number of loadcycles: The ‘N_jump’ [18] stands for cycle number thatmay be a predefined constant value or a number calculatedin-situ. No fatigue damage will be evaluated in side thisamount of load cycles but only at the end of it. With a pre-defined ‘N_jump’, all the Gaussian points continuously expe-rience the fatigue weakening effect. Governed by the inputS–N curves, this weakening effect is the function of theN_jump and local stresses states. The fatigue weakening

load cycle number: (a) CET3K and (b) CET12K.

Fig. 7. Normalised fatigue strengths vs logarithmic load cycle number: (a) CET3K and (b) CET12K. (c) S–N plots for CET3K and CET12K samples under fatigue loads of 75%, 80%and 85% of static strengths, respectively.

Fig. 8. A unit cell of the proposed meso-FE fatigue model: the impregnated tows are taken as piecewise UD composites; the S–N curves of UDs’ are fed to predict the fatigueproperties of a textile composite material.

J. Xu et al. / Composite Structures 135 (2016) 306–315 311

effects are cumulated by Palmgren–Miner’s rule – Eq. (8).Some Gaussian points are identified as damaged and theirstiffness is degraded by the anisotropic damage model; therest of the Gaussian points keep their stiffness unimpaired.The matrix material is taken as isotropic material, and thePalmgren–Miner’s rule and the multi-axial fatigue theory[19,20] – Eq. (9) are applied.

3.2. Geometrical model

The mesoscale model of textile composites has to be providedwith an adequate geometrical description of a unit cell, whichcan be created using a textile geometry generator [21,22] – Fig. 9(a) and converted thereafter to the FE mesh – Fig. 9(b) assistedby a FE mesher [23,24]. The FE-mesh will be eventually convertedto ABAQUS environment with the appropriate fibre orientation andPBCs – Fig. 9(c).

3.3. Fatigue failure criteria

On the one hand, the on-axis fatigue damage should be consid-ered separately because it is governed by breakage of the fibresother than inter-fibre matrix and fibre–matrix interface. On theother hand, the fibre breakage is the primary failure that willdegrade all the stiffness components by a factor of 0.01. Duringthe damage propagation, the local stress amplitudes keep changingdue to the stress redistribution introduced by the consecutive stiff-ness degradation of the neighbouring Gaussian points. In thismodel, the along-fibre fatigue damage is accumulated usingPalmgren–Miner linear damage rule, which yields the residual por-tion of the fatigue strength of a Gaussian point after a series ofblock loadings: r1

1;r21; . . . ;ri

1 with the correspondent cycle number

N1block;N

2block; . . . ;N

iblock – Fig. 10.

Rj1 ¼

Xj

k¼1

Nkblock

Nk1ðrk

1Þ;

Rs j1 ¼ 1� Rj1

ð8Þ

where Rj1 is the portion of exhausted strength and Nk

1ðrk1Þ is the

maximal number of cycle to failure on stress level rk1. The fraction

Nkblock=N

k1ðrk

1Þ is called ‘damage fraction’ for the k-th cycle block.For the matrix-dominated directions, Liu’s theory [19], which

accounts for multi-axial fatigue and independent to the materialtype, is applied. Consider a UD, the material coordinate system‘1–2–3’ is defined as such: the coordinate system obeys theright-hand rule; axis 1 follows the fibre direction; axis 2 is parallelto the unit cell plane; axis 3 is perpendicular to the plane 1–o–2following the right-hand-rule – Fig. 11.

In the material coordinate system, the ‘crack plane’ and ‘criticalplane’, based on which Liu came up with his multi-axial fatiguefailure criterion, have to be interpreted. The ‘crack plane’ is amicro-level fracture plane parallel to the fibres, whose normalhas an angle (hcrack) with axis 3 – Fig. 11. In this model, the poten-tial crack plane, a plane experiencing the maximum principalstress, will be numerically sought out. The ‘critical plane’ can betaken as a coordinate, into which the stresses are transformedand the fatigue damage is evaluated using Eq. (9). Based on thenotions of ‘crack plane’ and ‘critical plane’, Liu and Mahadevan[19] proposes a criterion for planar stress states – Eq. (9):ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rc

f ðNÞ� �2

þ sctðNÞ

� �2

þ krH

c

f ðNÞ� �2

s¼ b ð9Þ

This criterion is then extended to composite materials [20]. Thismodel is a second order combination of the peak values (magni-tudes) of cyclic normal stress rc , shear stress sc – and hydrostaticstress rH

c (mean normal stress) on the critical plane (subscript ‘c’

Fig. 9. Meso-FE model of textile composites: (a) twill weave geometry by WiseTex [22], (b) mesh generated by MeshTex, (c) FE model with matrix material in ABAQUS�.

Fig. 10. Application of Palmgren–Miner’s linear damage rule to a Gaussian point infibre direction: constant amplitude loading blocks.

Fig. 11. Illustration of critical plane and crack plane in the material coordinatesystem. The angles between crack plane and axis 3 (0

�< hcrack < 90

�), and the angle

between crack plane and critical plane (a) are depicted.

Fig. 12. Damage modes and the corresponding damage tensor [26]. D1, D2 and

312 J. Xu et al. / Composite Structures 135 (2016) 306–315

refers to ‘critical plane’). f ðNÞ and tðNÞ are tension–tension andshear fatigue strengths at load cycle N, represented by the inputS–N curves of UD composite. k and b are material parameterswhich can be determined by uniaxial and torsional fatigue tests.Based on Eq. (9), a 3-D failure criterion that accounts for damageaccumulation using Miner’s rule was proposed [12].

3.4. Anisotropic damage model

The post-damage stiffness degradation is described by theanisotropic damage model [25] – Fig. 12. The coordinate system1–2–3 is the material coordinate system in an impregnated fibrebundle. Damage mode 1 delineates fibre rupture while the modes2–4 are for inter-fibre cracks. For each of the damage modes thereis a corresponding damage tensor.

D ¼Xi

Di~ni �~ni ¼D1 0 00 D2 00 0 D3

264

375 i ¼ 1;2;3 ð10Þ

where Di is the principle value and ~ni is a unit vector. The physicalmeaning of the damage factor Di is the effective area reductioncaused by cracks. In calculation, Di are zero for a virgin materialand 0.99 for damage to avoid the numerical singularity. Using Di,the degraded stiffness matrix in post-damage phase can be calcu-lated as follows:

r1

r2

r3

s23

s31

s12

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

¼

d21Q11 d1d2Q12 d1d3Q13 0 0 0

d22Q22 d2d3Q23 0 0 0

d23Q33 0 0 0

d23Q44 0 0

Sym d31Q55 0

d12Q66

26666666666666664

37777777777777775

e1

e2

e3

c23

c31

c12

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

;

ð11Þwhere d and D have the following relations,

D3 are principle values of damage tensor, whose application is given in [1].

Fig. 13. Comparison of normalised S–N plot: (a) CET3K and (b) CET12K.

Fig. 14. Stress concentration introduced by crimp in warp tows: (a) a profile ofimpregnated twill weave preform in the composite; (b) bending effect and stressconcentration at location I.

Fig. 15. Predicted locations of fibre rupture in twill weave composite: (a) a unit cell of twlocations in the warp tows.

J. Xu et al. / Composite Structures 135 (2016) 306–315 313

d1 ¼ 1� D1; d2 ¼ 1� D2; d3 ¼ 1� D3

d23 ¼ 2d2d3d2þd3

� �2; d31 ¼ 2d3d1

d3þd1

� �2; d12 ¼ 2d1d2

d1þd2

� �2

8<: ð12Þ

4. Comparison and discussion

4.1. Comparison of S–N plot

The experimental moduli-life curves are used to determine thecatastrophic failure of the materials. As shown in Fig. 4(b), theaveraged moduli right before failure for CET3K and CET12K are85% and 75% of the static moduli, respectively. The calculation willterminate once the fatigue moduli drop to the 85% and 75% forCET3K and CET12K, respectively.

Based on the moduli-life curves, the predicted normalised fati-gue strengths of the two types of twill weave composites are pre-sented in Fig. 13(a) and Fig. 13(b) (solid symbols) together withthe experimental data (hollow symbols). In these figures, the fatiguestrengths are compared at the maximum fatigue load level of 75%,80% and 85% of the static strength. Reasonably good agreementshave been obtained for CET12K. For CET3K, the predicted fatiguestrengths tend to be slightly higher than the test data. However,

ill weave composite (matrix elements are set invisible), (b) predicted fibre rupture

Fig. 16. Stress contour of a11 in the warp tows of CET3K and CET12K. The applied external stress is 816 MPa, 85% of the strength for CET3K. (a) r11 contour for CET3K (justbefore fibre rupture at cycle number N = 104.5) with the location where r11 has maximum 2885 MPa. (b) r11 contour for CET12K (at the cycle number N = 104.5) with thelocation where r11 has maximum 2835 MPa.

Fig. 17. Fatigue life estimation by using the normalised maximum fatigue stressand the S–N curve of a UD composite (input data) before fibre rupture.

314 J. Xu et al. / Composite Structures 135 (2016) 306–315

comparing to the large experimental scatters in high stress levelfatigue tests, these deviations still fall into a reasonable range.

4.2. Fibre rupture in warp tows

Using the FE model, the locations of fibre rupture in the warptow can be predicted. As shown in Fig. 15, CET3K and CET12K exhi-bit the same fibre rupture locations. The fibre rupture will initiateat the location where (1) warp tow and weft tow have a contactsurface (in between the warp and weft tows at the cross points),and (2) the warp tow has the highest curvature [1] – Fig. 14(a).Fig. 14(b) shows the bending effect and where stress concentrationtakes place – location I.

In the following numerical tests, the same fatigue loadings of816 MPa, 85% of the static strength of CET3K, are applied to CET3Kand CET12K, respectively. The predicted first fibre ruptures are atload cycle number N ¼ 104:5 and N ¼ 105:25 respectively for CET3Kand CET12K. In Fig. 16 the r11 contours for CET3K and CET12K atcycle number N ¼ 104:5 are comparatively studied. Fig. 16(a) isthe contour for CET3K, just before the fibre rupture. CET12K expe-riences the same fatigue loading and number of cycles withoutfibre rupture – Fig. 16(b). The maximum fatigue stress (stress

concentration) in CET3K is 2885 MPa, 50 MPa higher than that inCET12K (2835 MPa).

The strengths of the impregnated tows can be calculated byusing the Chamis’ equations [27] and the tow VF for CET3K(67.2%) and CET12K (71.4%) – [1]. They are 3265 MPa and3500 MPa, respectively. Thus the normalised maximum stressesat the stress concentration locations for CET3K and CET12K inFig. 16 are 88% and 81%, respectively. The fatigue life can beroughly estimated by using the input S–N curve shown in Fig. 2(a). In Fig. 17, CET12K has lower normalised maximum fatiguestresses (81%) at the stress concentration locations than CET3K(88%) and hence postponed fibre ruptures (105.2 cycles vs 104.4

cycles) in the warp tows.

5. Conclusion

Two types of twill weave carbon fibre epoxy composite materi-als have been experimentally and numerically investigated undertension–tension fatigue loadings (85%, 80% and 75% of the staticstrength). They have the same fibre/resin system but differentgeometry of reinforcement (tow size and crimp). With the helpof DIC, the elongation of the samples is calculated by subtractingthe machine/grip deformation. Based on the sample elongation,the displacement AI and the fatigue moduli are obtained as wellas the S–N plots. From the experimental data, we conclude that:

1. CET12K has higher (18% higher) fatigue strengths as well as sta-tic strength compared to CET3K.

2. The displacement AI after the fatigue tests for CET12K is aver-aged at 0.45 mm, which is much higher than that of CET3K,0.18 mm – Fig. 4; the modulus for CET12K immediately beforecatastrophic failure has decreased to 75% of the virgin modulusand this value for CET3K is 85% – Fig. 4(b). One reason could bethe severe fatigue damage during the fatigue for CET12K –Fig. 5.

3. The curves of displacement AI vs fatigue life are divided intotwo groups by the material types. They tend to be dependenton the material types but irrelative to the load level.

4. The model is able to predict the materials’ fatigue lives and thefibre rupture locations that are introduced by stress concentra-tion. Higher crimp in CET3K introduces higher stress concentra-tion and leads to shorter fatigue lives.

J. Xu et al. / Composite Structures 135 (2016) 306–315 315

As a summary, CET12K shows much better fatigue resistancethan CET3K due to the flatter tow and lower stress concentrationin warp tows. These phenomena can be well explained using theproposed fatigue model [12]. Nevertheless, a model with delami-nation is necessary and more studies have to be carried out onstochastic breakage of the fibres in tows during fatigue.

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